20. Rheology & Linear Elasticity

I Main Topics A Rheology: Macroscopic deformation behavior B Linear elasticity for homogeneous isotropic materials

10/29/18 GG303 1 20. Rheology & Linear Elasticity Viscous (fluid) Behavior

http://manoa.hawaii.edu/graduate/content/slide-lava 10/29/18 GG303 2 20. Rheology & Linear Elasticity

Ductile (plastic) Behavior

http://www.hilo.hawaii.edu/~csav/gallery/scientists/LavaHammerL.jpg http://hvo.wr.usgs.gov/kilauea/update/images.html

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Elastic Behavior

https://thegeosphere.pbworks.com/w/page/24663884/Sumatra http://www.earth.ox.ac.uk/__data/assets/image/0006/3021/seismic_hammer.jpg

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Brittle Behavior (fracture)

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II Rheology: Macroscopic deformation behavior A Elasticity 1 Deformation is reversible when load is removed 2 Stress (σ) is related to strain (ε) 3 Deformation is not time dependent if load is constant 4 Examples: Seismic (acoustic) waves, http://www.fordogtrainers.com rubber ball

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II Rheology: Macroscopic deformation behavior A Elasticity 1 Deformation is reversible when load is removed 2 Stress (σ) is related to strain (ε) 3 Deformation is not time dependent if load is constant 4 Examples: Seismic (acoustic) waves, rubber ball

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II Rheology: Macroscopic deformation behavior B Viscosity 1 Deformation is irreversible when load is removed 2 Stress (σ) is related to strain rate (ε ! ) 3 Deformation is time dependent if load is constant 4 Examples: Lava flows, corn syrup http://wholefoodrecipes.net

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II Rheology: Macroscopic deformation behavior B Viscosity 1 Deformation is irreversible when load is removed 2 Stress (σ) is related to strain rate (ε ! ) 3 Deformation is time dependent if load is constant 4 Examples: Lava flows, corn syrup

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II Rheology: Macroscopic deformation behavior C Plasticity 1 No deformation until yield strength is locally exceeded; then irreversible deformation occurs under a constant load 2 Deformation can increase with time under a constant load 3 Examples: plastics, soils http://www.therapyputty.com/images/stretch6.jpg

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II Rheology: Macroscopic deformation behavior C Brittle Deformation 1 Discontinuous deformation 2 Failure surfaces separate

http://www.thefeeherytheory.com

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II Rheology: Macroscopic deformation behavior D Elasto-plastic rheology

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II Rheology: Macroscopic deformation behavior E Visco-plastic rheology

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II Rheology: Macroscopic deformation behavior F Power-law creep n (−Q/RT) 1 ε!= (σ1 − σ3) e 2 Example: rock salt

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II Rheology: Macroscopic deformation behavior G Linear vs. nonlinear behavior

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II Rheology: Macroscopic deformation behavior

H Rheology=f (σij ,fluid pressure, strain rate, chemistry, temperature) I Rheologic equation of real rocks = ?

10/29/18 GG303 16 20. Rheology & Linear Elasticity II Rheology (cont.) J Experimental results Axial stress Plastic

Elastic Increasing confining pressure Pc = confining pressure

Compression test data on Tennessee marble II from Wawersik and Fairhurst, 1970

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III Linear elasticity x A Force and displacement F of a spring (from Hooke, 1676): F= kx 1 F = force 2 k = spring constant k Dimensions:F/L F 3 x = displacement Dimensions: length L) x

10/29/18 GG303 18 20. Rheology & Linear Elasticity σ III Linear elasticity (cont.) B Hooke’s Law for L0+ΔL L0 uniaxial stress: σ = Eε ε=ΔL/L0 1 σ = uniaxial stress 2 E = Young’s modulus Dimensions: stress E 3 ε = strain σ (elongation) Dimensionless ε

10/29/18 GG303 19 20. Rheology & Linear Elasticity Typical rock moduli and strengths

Young’s Poisson’s Modulus Ratio (GPa) Uniaxial Strengths (MPa)

Rock type Emin Emax νmin νmax Rock type Tensile Tensile Comp. Comp. (low) (high) (low) (high) Quartzite 70 105 0.11 0.25 Quartzite 17 28 200 304

Gneiss 16 103 0.10 0.40 Gneiss 3 21 73 340

Basalt 16 101 0.13 0.38 Basalt 2 28 42 355

Granite 10 74 0.10 0.39 Granite 3 39 30 324

Limestone 1 92 0.08 0.39 Limestone 2 40 48 210

Sandstone 10 46 0.10 0.40 Sandstone 3 7 40 179

Shale 10 44 0.10 0.19 Shale 2 5 36 172

Coal 1.5 3.7 0.33 0.37 Coal 1.9 3.2 14 30

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σ1 III Linear elasticity (cont.) B Hooke’s Law for uniaxial L +ΔL stress (cont.): ε1 = σ1/E 0 L0 ε=ΔL/L0 1 σ2 = σ3 = 0

2 ε2 = ε3 = -νε1 a ν = Poisson’s ratio b ν is dimensionless c Strain in one direction tends to induce strain in another direction

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σ1 III Linear elasticity (cont.)

C Linear elasticity in 3D for L +ΔL 0 L0 homogeneous isotropic ε=ΔL/L0 materials By superposition:

1 εxx = σxx/E – (σyy+σzz)(ν/E)

2 εyy = σyy/E – (σzz+σxx)(ν/E)

3 εzz = σzz/E – (σxx+σyy)(ν/E)

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σ1 III Linear elasticity (cont.) C Linear elasticity in 3D for homogeneous isotropic L0+ΔL L0 ε=ΔL/L materials (cont.) 0 4 Directions of principal stresses and principal strains coincide 5 Extension in one direction can occur without tension 6 Compression in one direction can occur without shortening

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III Linear elasticity E Special cases 1 Isotropic (hydrostatic) stress

a σ1 = σ2 = σ3 b No shear stress 2 Uniaxial strain x a εxx= ε1≠ 0

b εyy = εzz = 0

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III Linear elasticity E Special cases 3 Plane stress (2D) σz = 0 “Thin plate” case 4 Plane strain (2D) εz = 0 a Displacement in z- direction is constant (e.g., zero) b Plate is confined between rigid walls c “Thick plate” case 10/29/18 GG303 25 20. Rheology & Linear Elasticity

III Linear elasticity E Special cases 5 Pure shear stress (2D)

σxx= -σyy; σzz=0

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III Linear elasticity

F Strain energy (W0) for uniaxial stress σxxdydz

1

2 W = (1/2)(σxxdydz) (εxxdx) ε dx 3 W = (1/2) (σxxεxx) (dxdydz) xx

4 W0 = W/(dxdydz) W0 = strain energy density

5 W0 = (1/2)(σxxεxx) 10/29/18 GG303 27 20. Rheology & Linear Elasticity

IIILinear elasticity

G Strain energy (W0) in 3D σxxdydz W0 = (1/2)(σ1ε1+σ2ε2+σ3ε3)

εxxdx

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III Linear elasticity 4 β = compressibility D Relationships among β = 1/K different elastic moduli ∆ = εxx+εyy+ εzz= -p/K 1 G = μ = shear modulus p = pressure G = E/(2[1+ν]) 5 P-wave speed: Vp ε = σ /2G xy xy ⎛ 4 ⎞ 2 λ = Lame' constant Vp = K + µ ρ ⎝⎜ 3 ⎠⎟ λ = Ev/([1 + ν][1 - 2ν]) 6 S-wave speed: Vs 3 K = bulk modulus K = E/(3[1 - 2ν])

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